Manchester Memoirs, W^I. Iviii. (1914), No. 5. 15 



I propose now to show that we are prepared to 

 deal with the cases where no identity of the stresses is 

 postulated. 



Returning to the equations numbered from (i) to (5), 

 and in (3), write 



^_ S-(^, + ;//,) 



vxcz 



u^m^±^ (.5), 



cxcy 



while the relations (6) are retained, namely, 



yir^ = ;,/(e^ + a,) etc. 



We thus obtain 



djoz 



exes 

 17=U' + A- (26). 



CXCV 



Also we find P, Q, and R in the form of (27) which will 

 replace (7), that is 



/>= „e, - ;/U^(ei + e,) + .-..(e, + 93) - ^- - ^ , 



Key cz- ) vj c^ 



etc., (27), 



while the forms of P', Q', and K' in (8) are unchanged. 



This section is not intended to treat of the case when 

 the stresses are functions of higher powers of the elements 

 of strain. That case presents points of difference and 

 will be dealt with in a subsequent Part. 



